Finite Math Examples

$- 0.8 x (x - 9) = 1.4 x + 5.3$

Step 1

Simplify $- 0.8 x (x - 9)$ .

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Step 1.1

Rewrite.

$0 + 0 - 0.8 x (x - 9) = 1.4 x + 5.3$

Step 1.2

Simplify by adding zeros.

$- 0.8 x (x - 9) = 1.4 x + 5.3$

Step 1.3

Apply the distributive property.

$- 0.8 x \cdot x - 0.8 x \cdot - 9 = 1.4 x + 5.3$

Step 1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1

Move $x$ .

$- 0.8 (x \cdot x) - 0.8 x \cdot - 9 = 1.4 x + 5.3$

Step 1.4.2

Multiply $x$ by $x$ .

$- 0.8 x^{2} - 0.8 x \cdot - 9 = 1.4 x + 5.3$

Step 1.5

Multiply $- 9$ by $- 0.8$ .

$- 0.8 x^{2} + 7.2 x = 1.4 x + 5.3$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $1.4 x$ from both sides of the equation.

$- 0.8 x^{2} + 7.2 x - 1.4 x = 5.3$

Step 2.2

Subtract $1.4 x$ from $7.2 x$ .

$- 0.8 x^{2} + 5.8 x = 5.3$

Step 3

Subtract $5.3$ from both sides of the equation.

$- 0.8 x^{2} + 5.8 x - 5.3 = 0$

Step 4

Factor $- 0.1$ out of $- 0.8 x^{2} + 5.8 x - 5.3$ .

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Step 4.1

Factor $- 0.1$ out of $- 0.8 x^{2}$ .

$- 0.1 (8 x^{2}) + 5.8 x - 5.3 = 0$

Step 4.2

Factor $- 0.1$ out of $5.8 x$ .

$- 0.1 (8 x^{2}) - 0.1 (- 58 x) - 5.3 = 0$

Step 4.3

Factor $- 0.1$ out of $- 5.3$ .

$- 0.1 (8 x^{2}) - 0.1 (- 58 x) - 0.1 \cdot 53 = 0$

Step 4.4

Factor $- 0.1$ out of $- 0.1 (8 x^{2}) - 0.1 (- 58 x)$ .

$- 0.1 (8 x^{2} - 58 x) - 0.1 \cdot 53 = 0$

Step 4.5

Factor $- 0.1$ out of $- 0.1 (8 x^{2} - 58 x) - 0.1 (53)$ .

$- 0.1 (8 x^{2} - 58 x + 53) = 0$

Step 5

Divide each term in $- 0.1 (8 x^{2} - 58 x + 53) = 0$ by $- 0.1$ and simplify.

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Step 5.1

Divide each term in $- 0.1 (8 x^{2} - 58 x + 53) = 0$ by $- 0.1$ .

$\frac{- 0.1 (8 x^{2} - 58 x + 53)}{- 0.1} = \frac{0}{- 0.1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 0.1$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 0.1 (8 x^{2} - 58 x + 53)}{- 0.1} = \frac{0}{- 0.1}$

Step 5.2.1.2

Divide $8 x^{2} - 58 x + 53$ by $1$ .

$8 x^{2} - 58 x + 53 = \frac{0}{- 0.1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $- 0.1$ .

$8 x^{2} - 58 x + 53 = 0$

Step 6

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 7

Substitute the values $a = 8$ , $b = - 58$ , and $c = 53$ into the quadratic formula and solve for $x$ .

$\frac{58 \pm \sqrt{{(- 58)}^{2} - 4 \cdot (8 \cdot 53)}}{2 \cdot 8}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 58$ to the power of $2$ .

$x = \frac{58 \pm \sqrt{3364 - 4 \cdot 8 \cdot 53}}{2 \cdot 8}$

Step 8.1.2

Multiply $- 4 \cdot 8 \cdot 53$ .

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Step 8.1.2.1

Multiply $- 4$ by $8$ .

$x = \frac{58 \pm \sqrt{3364 - 32 \cdot 53}}{2 \cdot 8}$

Step 8.1.2.2

Multiply $- 32$ by $53$ .

$x = \frac{58 \pm \sqrt{3364 - 1696}}{2 \cdot 8}$

Step 8.1.3

Subtract $1696$ from $3364$ .

$x = \frac{58 \pm \sqrt{1668}}{2 \cdot 8}$

Step 8.1.4

Rewrite $1668$ as $2^{2} \cdot 417$ .

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Step 8.1.4.1

Factor $4$ out of $1668$ .

$x = \frac{58 \pm \sqrt{4 (417)}}{2 \cdot 8}$

Step 8.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{58 \pm \sqrt{2^{2} \cdot 417}}{2 \cdot 8}$

Step 8.1.5

Pull terms out from under the radical.

$x = \frac{58 \pm 2 \sqrt{417}}{2 \cdot 8}$

Step 8.2

Multiply $2$ by $8$ .

$x = \frac{58 \pm 2 \sqrt{417}}{16}$

Step 8.3

Simplify $\frac{58 \pm 2 \sqrt{417}}{16}$ .

$x = \frac{29 \pm \sqrt{417}}{8}$

Step 9

The final answer is the combination of both solutions.

$x = \frac{29 + \sqrt{417}}{8}, \frac{29 - \sqrt{417}}{8}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{29 + \sqrt{417}}{8}, \frac{29 - \sqrt{417}}{8}$

Decimal Form:

$x = 6.17757223 \dots, 1.07242776 \dots$